MFT cameras don't share the same wide field of view as a full frame DSLR. In addition the 4K Video setting on the Panasonic DMC-GH4 is further cropped in when compared to it's own 1080p setting.
One benefit to shooting 4K video is the extra resolution that will allow you to recompose, level a shot, or even zoom in to frame when outputting to 1080p. Seeing how my Rokinon 7.5mm Fisheye lens became less 'fisheye-like' with the 4K crop, I was also curious about using the extra resolution to correct the remaining distortion and finally outputting to 1080p. Nothing new as many users have 'de-fished' the Ultra Wide look from early GoPro cameras. I attempted to DeFish a BlackMagic Pocket Cinema Camera Clip with my fisheye lens that ended up looking great, but 1080p can only be stretched so far.
My tests with the GH4 4K Video show that the image is far from perfect, but far better when attempting to defish from a 1080p file. There's an obvious loss of sharpness towards the edges if you're really nit picky, but in many cases where you have to work in very tight situations with a GH4, taking advantage of the extra resolution in 4K is a good option. (BTW no sharpening has been done in post in this video)